# signature

 $\Sigma:=\left(\bigcup_{n\in\omega}\mathcal{R}_{n}\right)\cup\left(\bigcup_{n% \in\omega}\mathcal{F}_{n}\right)\cup\mathcal{C}$

where for each natural number  $n>0$,

$\bullet$

$\mathcal{R}_{n}$ is a (usually countable  ) set of $n$-ary relation symbols.

$\bullet$

$\mathcal{F}_{n}$ is a (usually countable) set of $n$-ary function symbols.

$\bullet$

$\mathcal{C}$ is a (usually countable) set of constant symbols.

We require that all these sets be pairwise disjoint.

Given a signature $\Sigma$, a $\Sigma$-structure  is then a structure $\mathcal{A}$, whose underlying set is some set $A$, with elements $\mathcal{A}_{c}\in A$ for each constant symbol $c\in\Sigma$, $n$-ary operations  $\mathcal{A}_{f}$ on $A$ for each $n$-ary function symbol $f\in\Sigma$, for each $n$, and $m$-ary relations   $\mathcal{A}_{R}$ on $A$ for each $m$-ary relation symbol $R\in\Sigma$.

On the other hand, every structure is associated with a signature. For every structure, it has an underlying set, together with a collection  of “designated” objects that “define” the structure. These objects may be elements of the underlying set, operations on the set, or relations on the set. For each such “designated” object, pick a symbol for it. Make sure all symbols used are distinct from one another. Then the collection of all such symbols is a signature for the structure.

For most structures that we encounter, the set $\Sigma$ is finite, but we allow it to be infinite  , even uncountable, as this sometimes makes things easier, and just about everything still works when the signature is uncountable.

Examples:

Remark. Given a signature $\Sigma$, the set $L$ of logical symbols from first order logic, and a countably infinite  set $V$ of variables, we can form a first order language, consisting of all formulas  built from these symbols (in $\Sigma\cup L\cup V$). The language  so-created is uniquely determined by $\Sigma$. In the literature, it is a common practice to identify $\Sigma$ both as a signature and the unique language it generates.

## References

• 1 W. Hodges, , Cambridge University Press, (1997).
• 2 D. Marker, Model Theory, An Introduction, Springer, (2002).
Title signature Signature 2013-03-22 13:51:48 2013-03-22 13:51:48 CWoo (3771) CWoo (3771) 16 CWoo (3771) Definition msc 03C07 language non-logical symbols constant symbol function symbol relation symbol